satisfying certain conditions (prop. 6.1.2.6) which are such that if C0=*C_0 = {*} is the point we have an internal group in a homotopical sense, given by an object C1C_1 equipped with a coherently associative multiplication operation C1×C1→C1C_1 \times C_1 \to C_1 generalizing that of StasheffH-space from the (∞,1)(\infty,1)-category Top to arbitrary (∞,1)(\infty,1)-categories.

is just the identity map, using that Obj(G)=GObj(G) = G and Mor(BG)=GMor(\mathbf{B}G) = G.

Deloopings of higher categorical structures

There is also a notion of delooping which takes a pointed (n,k+1)(n, k+1)-category CC to a pointed (n+1,k)(n+1, k)-category BC\mathbf{B} C in which BC\mathbf{B} C has a single 00-cell •\bullet, and where hom(•,•)=C\hom(\bullet, \bullet) = C. This is a tautological construction if one accepts the delooping hypothesis, which views a (n,k+1)(n, k+1)-category CC as a special type of (n+k+1)(n+k+1)-category, namely a pointed kk-connected (n+k+1)(n+k+1)-category: by viewing such as a fortiori a pointed (k−1)(k-1)-connected (n+k+1)(n+k+1)-category, we get the delooping BC\mathbf{B} C.

This is just a generalization of the fact that a monoidMM gives rise to a one-object category (which we are denoting BM\mathbf{B} M). For an important example: a monoidal categoryMM has an associated delooping bicategoryBM\mathbf{B} M, where

BM\mathbf{B} M has a single 00-cell •\bullet,

the 11-cells •→•\bullet \to \bullet of BM\mathbf{B} M are named by objects of MM, and the composite of •→a•→b•\bullet \stackrel{a}{\to} \bullet \stackrel{b}{\to} \bullet is •→a⊗b•\bullet \stackrel{a \otimes b}{\to} \bullet (using the monoidal product ⊗\otimes of MM),

the 22-cells of BM\mathbf{B} M are similarly named by morphisms of MM; the vertical composition of 22-cells in BM\mathbf{B} M is given by composition of morphisms of MM, and the horizontal composition of 22-cells in BM\mathbf{B} M is given by taking the monoidal product of the morphisms that name them in MM.